#27532878

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Splittings

Sylow

C₅₃ ⋊ #32901

Subgroups

	subgroup	G/H
center	C₂²	C₂×#13133186
commutator, cosocle	C₂⁴×C₅₃	C₂⁴
Frattini	C₂⁴	C₂³×D₅₃
Fitting, prime radical	C₂×C₅₃×#585	C₂
socle	C₂²×C₅₃	C₂×#120
hypercenter	C₂×#585	D₅₃
hypocenter	C₅₃	#32901
LCS	G ▷ C₂⁴×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#585

3706 conjugacy classes of subgroups; only the 315 normal subgroups are shown.

normal subgroups only

This group has 3706 conjugacy classes of subgroups — too many to draw. Only its 315 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 73
transitive degree	μ_t(G) = 3392

A minimal faithful permutation representation — 4 generators on 73 points:

(1,2,4,6,8,10,12,14,16,18,20,22,24,26,28,30,32,34,36,38,40,42,44,46,48,50,52,53,51,49,47,45,43,41,39,37,35,33,31,29,27,25,23,21,19,17,15,13,11,9,7,5,3)(55,57)(58,62,63,59)(60,67,68,64)(61,69,70,65)(66,72,73,71);(58,61)(59,65)(60,66)(62,69)(63,70)(64,71)(67,72)(68,73);(58,60)(59,64)(61,66)(62,67)(63,68)(65,71)(69,72)(70,73);(2,3)(4,5)(6,7)(8,9)(10,11)(12,13)(14,15)(16,17)(18,19)(20,21)(22,23)(24,25)(26,27)(28,29)(30,31)(32,33)(34,35)(36,37)(38,39)(40,41)(42,43)(44,45)(46,47)(48,49)(50,51)(52,53)(54,55)(56,57)(58,59,63,62)(60,64,68,67)(61,65,70,69)(66,72,73,71)

Polycyclic presentation:

2^8,53|a^2=h,d^2=h,b^a=be,c^a=cf,d^a=dg,i^a=i^52,f^b=fh,e^c=eh

Conjugacy classes & character table

1817 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 318354);